Let 𝑋1,…,𝑋𝑛 be i.i.d. random variables with distribution (𝜃,𝜃) , for some unknown parameter 𝜃>0 .

Question

Find an interval I𝜃 (that depends on 𝜃 ) centered about 𝑋⎯⎯⎯⎯⎯𝑛 such that

P(I𝜃 <- 𝜃 )=0.9for all 𝑛(i.e, not only for large 𝑛).

(Write barX_n for 𝑋⎯⎯⎯⎯⎯𝑛 . Use the estimate 𝑞0.05≈1.6448 for best results.)

I𝜃=[A𝜃, B𝜃] for
A𝜃=?
B𝜃=?

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Again, use the estimate 𝑞0.05≈1.6448 for best results.

Now, find a confidence interval Iplug-in with asymptotic confidence level 90% by plugging in 𝑋⎯⎯⎯⎯⎯𝑛 for all occurrences of 𝜃 in I𝜃 .

Iplug-in=[𝐴plug-in, 𝐵plug-in] for
𝐴plug-in=?
𝐵plug-in=?

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Finally, find a confidence interval Isolve for 𝜃 with nonasymptotic level 90% solving the bounds in I𝜃 for 𝜃 .

Isolve=[𝐴solve,𝐵solve] for
𝐴solve=?
𝐵solve=?

Answer 1

clarification : Let 𝑋1,…,𝑋𝑛 be i.i.d. random variables with distribution N(𝜃,𝜃) , for some unknown parameter 𝜃>0 .

Answer 2

a) True
b) theta and variance = theta / n
c) barX_n - 1.6448*sqrt(theta/n)
d) barX_n - 1.6448*sqrt(barX_n/n)